Theorem nmoofval 31346

Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmoofval.1	⊢ 𝑋 = (BaseSet‘𝑈)
nmoofval.2	⊢ 𝑌 = (BaseSet‘𝑊)
nmoofval.3	⊢ 𝐿 = (normCV‘𝑈)
nmoofval.4	⊢ 𝑀 = (normCV‘𝑊)
nmoofval.6	⊢ 𝑁 = (𝑈 normOpOLD 𝑊)

Assertion

Ref	Expression
nmoofval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))

Distinct variable groups:   𝑥,𝑡,𝑧,𝑈   𝑡,𝑊,𝑥,𝑧   𝑡,𝑋,𝑧   𝑡,𝑌,𝑥   𝑡,𝐿   𝑡,𝑀

Allowed substitution hints:   𝐿(𝑥, 𝑧)   𝑀(𝑥, 𝑧)   𝑁(𝑥, 𝑧, 𝑡)   𝑋(𝑥)   𝑌(𝑧)

Proof of Theorem nmoofval

Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmoofval.6	. 2 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊)
2		fveq2 6877	. . . . . 6 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		nmoofval.1	. . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	eqtr4di 2814	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5	4	oveq2d 7428	. . . 4 ⊢ (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑m 𝑋))
6		fveq2 6877	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
7		nmoofval.3	. . . . . . . . . . 11 ⊢ 𝐿 = (normCV‘𝑈)
8	6, 7	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = 𝐿)
9	8	fveq1d 6879	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢)‘𝑧) = (𝐿‘𝑧))
10	9	breq1d 5113	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (((normCV‘𝑢)‘𝑧) ≤ 1 ↔ (𝐿‘𝑧) ≤ 1))
11	10	anbi1d 643	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))) ↔ ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))))
12	4, 11	rexeqbidv 3336	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))) ↔ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))))
13	12	abbidv 2827	. . . . 5 ⊢ (𝑢 = 𝑈 → {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))} = {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))})
14	13	supeq1d 9422	. . . 4 ⊢ (𝑢 = 𝑈 → sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < ))
15	5, 14	mpteq12dv 5192	. . 3 ⊢ (𝑢 = 𝑈 → (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )) = (𝑡 ∈ ((BaseSet‘𝑤) ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))
16		fveq2 6877	. . . . . 6 ⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17		nmoofval.2	. . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊)
18	16, 17	eqtr4di 2814	. . . . 5 ⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
19	18	oveq1d 7427	. . . 4 ⊢ (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑m 𝑋) = (𝑌 ↑m 𝑋))
20		fveq2 6877	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊))
21		nmoofval.4	. . . . . . . . . . 11 ⊢ 𝑀 = (normCV‘𝑊)
22	20, 21	eqtr4di 2814	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀)
23	22	fveq1d 6879	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤)‘(𝑡‘𝑧)) = (𝑀‘(𝑡‘𝑧)))
24	23	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)) ↔ 𝑥 = (𝑀‘(𝑡‘𝑧))))
25	24	anbi2d 642	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))) ↔ ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))))
26	25	rexbidv 3187	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧))) ↔ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))))
27	26	abbidv 2827	. . . . 5 ⊢ (𝑤 = 𝑊 → {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))} = {𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))})
28	27	supeq1d 9422	. . . 4 ⊢ (𝑤 = 𝑊 → sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < ))
29	19, 28	mpteq12dv 5192	. . 3 ⊢ (𝑤 = 𝑊 → (𝑡 ∈ ((BaseSet‘𝑤) ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))
30		df-nmoo 31329	. . 3 ⊢ normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑m (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, < )))
31		ovex 7445	. . . 4 ⊢ (𝑌 ↑m 𝑋) ∈ V
32	31	mptex 7221	. . 3 ⊢ (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )) ∈ V
33	15, 29, 30, 32	ovmpo 7572	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 normOpOLD 𝑊) = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))
34	1, 33	eqtrid 2808	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌 ↑m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧 ∈ 𝑋 ((𝐿‘𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡‘𝑧)))}, ℝ*, < )))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∃wrex 3087   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831  supcsup 9416  1c1 11182  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  NrmCVeccnv 31168  BaseSetcba 31170  norm_CVcnmcv 31174   normOp_OLD cnmoo 31325

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-sup 9418  df-nmoo 31329

This theorem is used by:  nmooval  31347  hhnmoi  32485